Multivariate Granger Causality and Generalized Variance

TL;DR

This paper extends Granger causality analysis to multivariate sets of variables, providing a theoretically sound framework based on generalized variances, with applications demonstrated in neuroscience.

Contribution

It introduces a principled multivariate Granger causality framework using generalized variances, addressing limitations of univariate approaches and enabling analysis of interactions among variable groups.

Findings

Supports a comprehensive extension of Granger causality to multivariate data

Highlights advantages of generalized variance measure in causality analysis

Demonstrates applications in neuroscience revealing new functional relations

Abstract

Granger causality analysis is a popular method for inference on directed interactions in complex systems of many variables. A shortcoming of the standard framework for Granger causality is that it only allows for examination of interactions between single (univariate) variables within a system, perhaps conditioned on other variables. However, interactions do not necessarily take place between single variables, but may occur among groups, or "ensembles", of variables. In this study we establish a principled framework for Granger causality in the context of causal interactions among two or more multivariate sets of variables. Building on Geweke's seminal 1982 work, we offer new justifications for one particular form of multivariate Granger causality based on the generalized variances of residual errors. Taken together, our results support a comprehensive and theoretically consistent extension of Granger causality to the multivariate case. Treated individually, they highlight several specific advantages of the generalized variance measure, which we illustrate using applications in neuroscience as an example. We further show how the measure can be used to define "partial" Granger causality in the multivariate context and we also motivate reformulations of "causal density" and "Granger autonomy". Our results are directly applicable to experimental data and promise to reveal new types of functional relations in complex systems, neural and otherwise.